Arguably most of the nuclear tests were done to prevent war. Each side wanted to show the other they could match and one-up them in nuclear tech. Neither really wanted to ever actually use them to kill people.
Plenty of people on both sides really wanted to use them. Nobel Peace Prize winner, Kissinger, being one of them. Fortunately, cooler heads held the launch codes and we all prevailed.
I think a lot of people on both sides were prepared to use them. Fortunately only a very small number of people wanted to use them and were in any position to actually do so.
Other way around. We had the Trinity test, Hiroshima & Nagasaki, and then we set off the rest. Unless you meant "preparations" for killing more people with our nuclear weapons, in which case you may be correct.
The latter tests arguably weren't performed to improve the killing capabilities of the bombs but rather to understand the physics and physiological impacts of bomb blasts. Given that it's possible (albeit unlikely) we never see a nuclear weapon used in war again I don't see it as being completely obvious.
